Let us begin by repeating the definition of a Multinomial random variable.
Consider the bag of words model where we’re counting the nubmer of words in a
document, where the words are generated from a fixed dictionary. The
probability mass function (PMF) is defined as

where $\pi_i$ is the probability of $i-th$ word, $x_i$ is the nubmer of
occurences of that word, $m$ is the number of words in the dictionary, and
$n$ is the total number of occurences of all words.

Since the Multinomial distribution comes from the exponential family, we know
computing the log-likelihood will give us a simpler expression, and since
$\log$ is concave computing the MLE on the log-likelihood will be equivalent
as computing it on the original likelihood function.